Sine and Cosine Rules Higher

1 · Labelling any triangle Higher

You need right-angled trigonometry first (see Trigonometry and exact values, or the KS3 page Trigonometric ratios). For a general triangle we use a standard labelling:

• The corners (vertices) are capital letters A, B, C, and the angle at each corner uses the same letter.
• Each side gets the lower-case letter of the angle opposite it: side a is opposite angle A, b opposite B, c opposite C.

A side and its opposite angle make a pair (same colour below). Spotting pairs is the key to choosing a rule. Drag the corners: the biggest angle is always opposite the longest side.

2 · The sine rule: where it comes from Higher

Cut the triangle into two right-angled triangles by dropping a perpendicular (a line at 90°) from C to the line AB. Call its length h. Then write h in two ways, once from each right-angled triangle. Step through, and drag the corners to check that it works for every shape, even when the perpendicular lands outside the triangle.

You only ever use two of the three fractions at once. You need one complete pair (a side and its opposite angle) plus one more piece of information.

3 · Using the sine rule Higher

To find a side, put the unknown side on top: b/sin B = a/sin A, then multiply both sides by sin B. To find an angle, put the sines on top: sin B/b = sin A/a, multiply by b, then use sin⁻¹. If you're given two angles, the third one is 180° minus the other two; you might need it to make a pair.

4 · The ambiguous case Higher

Here's a surprise. Suppose you know angle A, side b and side a (which is opposite A). Picture side a as a stick hinged at C, swinging until it hits the base line. Sometimes it can land in two places, giving two different triangles that both fit the information. That's because sin x = sin(180° − x): for example sin 50° = sin 130°. So when the sine rule gives you an angle, there may be a second, obtuse answer: 180° minus the first.

5 · The cosine rule: Pythagoras with a correction Higher

If you know two sides and the angle between them, there's no complete pair, so the sine rule can't start. Pythagoras says c² = a² + b² when C = 90°. For other angles, c² needs a correction. Make C smaller than 90° and the triangle closes up, so c is shorter than Pythagoras says; make C bigger and c is longer. The correction is exactly 2ab cos C.

Practise both ways round. The commonest error is BIDMAS: in a² + b² − 2ab cos C, work out 2ab cos C first and subtract it. Don't do a² + b² − 2ab and then multiply by cos C.

6 · Area = ½ab sin C Higher

You know area = ½ × base × height. Take a as the base. The height is the perpendicular from A to the line BC, and in the right-angled triangle it makes with side b, height = b sin C. So

area = ½ × a × b sin C

Use it with two sides and the angle between them (the included angle). The letters can change, e.g. ½bc sin A; the pattern is "two sides and the angle where they meet". Drag the corners: both calculations always agree.

7 · Which rule? Higher

The hardest part of an exam question is often choosing. Work through the flowchart with a question in mind.

Now practise choosing. Each triangle shows what you know; the question mark is what you want.

8 · Bearings and real problems Higher

A ship sails from A to B, then turns and sails to C. How far is C from A, and on what bearing? (Bearings are measured clockwise from north as three figures; see the Bearings page.) The two legs and the angle between them at B give two sides and the included angle: a cosine rule question. The hard part is finding angle ABC, using the parallel north lines.

Worked examples

Quick check

What you've learned

• Label a general triangle with side a opposite angle A, and so on. A side with its opposite angle is a pair.
• Sine rule: a/sin A = b/sin B = c/sin C, proved by writing the height h = b sin A = a sin B. Use it when you know a complete pair. Unknown side on top for a side; sines on top for an angle.
• Ambiguous case: sin x = sin(180° − x), so an angle found with the sine rule may have a second, obtuse answer. Check whether it fits.
• Cosine rule: c² = a² + b² − 2ab cos C, Pythagoras with a correction. Use it for two sides and the included angle (find the third side), or three sides (find an angle with cos C = (a² + b² − c²) ÷ 2ab).
• Area of any triangle = ½ab sin C, from ½ × base × height with height b sin C.
• Choosing: right angle → SOH CAH TOA; complete pair → sine rule; otherwise → cosine rule; area → ½ab sin C.
• In bearings problems, use parallel north lines to find the angle inside the triangle, then apply the rules.